Elementary Linear Algebra, 11th Edition

CHAPTER 8General Linear Transformations

CHAPTER CONTENTS

1. 8.1 General Linear Transformations
2. 8.2 Compositions and Inverse Transformations
3. 8.3 Isomorphism
4. 8.4 Matrices for General Linear Transformations
5. 8.5 Similarity

INTRODUCTION

In earlier sections we studied linear transformations from $R^n$ to $R^m$. In this chapter we will define and study linear transformations from a general vector space $V$ to a general vector space $W$. The results we will obtain here have important applications in physics, engineering, and various branches of mathematics.

8.1 General Linear Transformations

Up to now our study of linear transformations has focused on transformations from $R^n$ to $R^m$. In this section we will turn our attention to linear transformations involving general vector spaces. We will illustrate ways in which such transformations arise, and we will establish a fundamental relationship between general $n$-dimensional vector spaces and $R^n$.

Definitions and Terminology

In Section 1.8 we defined a matrix transformation $T_A:R^n\to R^m$ to be a mapping of the form

$$T_A\left(\mathbf{x}\right)=A\mathbf{x}$$

in which $A$ is an $m\times n$ matrix. We subsequently established in Theorem 1.8.3 that the matrix transformations are precisely the linear transformations from $R^n$ to $R^m$, that is, the transformations with the linearity properties

$$\begin{array}{ccc}\hfill T\left(\mathbf{u}+\mathbf{v}\right)=T\left(\mathbf{u}\right)+T\left(\mathbf{v}\right)\hfill & \hfill \text{and}\hfill & \hfill T\left(k\mathbf{u}\right)=kT\left(\mathbf{u}\right)\hfill \end{array}$$

We will use these two properties as the starting point for defining more general linear transformations. ...